On the Belyi degree(s) of a curve defined over a number field

Leonardo Zapponi

Published 2009-04-06Version 1

Belyi's theorem asserts that a smooth projective curve $X$ defined over a number field can be realized as a cover of the projective line unramified outside three points. In this short paper we investigate the bejaviour of the minimal degree of such a cover. More precisely, we start by defining the absolute Belyi degree of X, which only depends on the $\bar{\bold Q}$-isomorphism class of $X$. We then give a lower bound of this invariant, only depending on the stable primes of bad reduction (as defined in the paper) and we show that this bound is sharp. In the second part of the paper, we introduce the relative Belyi degree of a curve X defined over a fixed number field $K$. We first prove that there exist finitely many $K$-isomorphism classes of curves of bounded (relative) Belyi degree and we then obtain a lower bound, only depending on the primes of bad reduction of the minimal regular model of $X$ over (the ring of integers of) $K$.